we mean the greatest lower bound for numbers \(\lambda \)  for which series (REF ) converges.
Note that if \(\lambda \)  equals to a convergent exponent then series (REF ) may or may not be convergent. For instance, the sequences \(a_{n}= 1/n^{\lambda }\)  and \(1/(n\ln ^{2} n)^{\lambda }\)  have the same convergent exponent \(\lambda =1,\)  but in the first case the series (REF ) is divergent when \(\lambda =1\)  while in the second one it is convergent. In this paper we have a special interest regarding the first case. Consider the following obvious relation between the convergence exponent \(\rho _{1}\)  and the genus \(p\)  of the corresponding canonical product \(p \le \rho _{1}\le p+1.\)  It is clear that if \(\rho _{1}\)  is integer, then \(p=\rho _{1},\)  when the series (REF ) diverges for \(\lambda =\rho _{1},\)  while \(\rho _{1}=p+1\)  means that the series converges (in accordance with the definition of \(p\) ). In the monograph [1]} it is considered a more precise characteristic of the density of the sequence \(\lbrace a_{n}\rbrace _{1}^{\infty }\)  than the convergence exponent. Thus, there is defined the so called growth of the function \(n(r)\)  equals to a number of points of the sequence in the circle \(|z|<r.\)  By upper density of the sequence we call a number
\(\Delta =\overline{\lim \limits _{r\rightarrow \infty }} n(t)/t^{\rho _{1}},\) 
